Through-the-wall-imaging (TWI) can detect objects in a scene behind a wall. In a typical application, a transmitting antenna array emits a radar pulse that propagates through the wall. The pulse is reflected by the objects and propagate back to a receiving antenna array as a set of echoes. The composition of the scene can then be visualized by generating a radar image that represents scene, including the number, locations and shape of the objects and reflectivities of the objects. However, depending on dielectric permittivity and permeability of the walls, the received signal is often corrupted with indirect secondary reflections due to the walls, which result in artifacts, such as ghosts, that clutter the radar image. Therefore, it is desired to reduction artifacts and significantly improve the quality and applicability of TWI.
Signal Model
A TWI radar imaging system typically includes arrays of one or more transmitting antennas and receiving antennas, (Ns), Nr. A time-domain waveform of the pulse transmitted by each source is s(t), and a primary impulse response, excluding multi-path reflections, of the scene is gp (t, nr, ns) at the receiving antenna nr ∈ {1, . . . Nr} as a reflection of a pulse from transmittng antenna ns ∈ {1, . . . Ns}. The impulse response of the multi-path reflections due to the wall clutter is gm(t, nr, ns).
Using a conventional time-domain representation, the received signal can be represented byr(t, nr, ns)=s(t)*(gp(t, nr, ns)+gm(t, nr, ns)),   (1)where * is a convolution operator.
Without loss of generality, if there are K objects in the scene, each inducing a primary impulse response gk(t, nr, ns) indexed by k ∈ {1 . . . K}. The impulse response can be modeled by a convolution of a delay kernel d(t, nr, ns) with the primary impulse response gk(t, nr, ns) of each object, such thatgp(t, nr, ns)=Σk=1K gk(t, nr, ns), andgm(t ,nr, ns)=d(t, nr, ns)*(Σk=1Kgk(t ,nr, ns)).
For a particular transmitter-receiver antenna pair (nr, ns), all primary reflections experience the same delay convolution kernel d(t, nr, ns) when generating the clutter. The delay kernel can be regarded as a weighted Dirac delta functiond(t)=Σjw(tj)δ(t−tj),where tj>0 is the time delay at which the reflections reach the receiving antenna from the jth multi-path source, and w(tj) is an attenuation weight of the jth path.
The definition of the sparse delay kernel d(t, nr, ns) can be extended to that of an activation function that generates both the primary and multiple impulse responses by allowing ti≧0. Consequently, the received signal can be expressed as a superposition of the primary responses of all K objects convolved with an activation function asr(t, nr, ns)=s(t)*Σk=1Kd(t, nr, ns)*gk(t, nr, ns),   (3)where d(t, nr, ns) is independent of k.
Kaczmarz Method
The Kaczmarz method can be used to determine a solution x of large overdetermined systems of linear equations Ax=r, where A ∈ M×N has full column rank and r ∈ M. The procedure sequentially cycles through the rows of A, orthogonally projecting the solution estimate at iteration j onto the solution space given by a row or block of rows Aj, such that
                              x          j                =                              x                          j              -              1                                +                                    A              j              H                        ⁢                                                                                r                    j                                    -                                      〈                                                                  A                        j                                            ,                                              x                                                  j                          -                          1                                                                                      〉                                                                                                                                  A                      j                                                                            2                  2                                            .                                                          (        4        )            
Randomizing the row selection criteria improves the convergence performance of the Kaczmarz method. A sparse randomized Kaczmarz (SRK) projects the iterate xj−1 onto a subset of rows of A weighted by a diagonal matrix Wj, i.e.
                              x          j                =                              x                          j              -              1                                +                                    W              j                        ⁢                          A              j              H                        ⁢                                                                                r                    j                                    -                                      〈                                                                                            A                          j                                                ⁢                                                  W                          j                                                                    ,                                              x                                                  j                          -                          1                                                                                      〉                                                                                                                                                          A                        j                                            ⁢                                              W                        j                                                                                                  2                  2                                            .                                                          (        5        )            
The weighting is based on identifying, in each iteration j, a support estimate Tj for x corresponding to the largest {circumflex over (k)} entries of the iterate xj, where {circumflex over (k)} is some predetermined sparsity level. The weighting gradually scales down the entries of Aj that lie outside of Tj by a weight ωj=1/√{square root over (j)}. As the number of
iterations becomes large, the weight
      1          j        ->  0and the method begins to resemble the randomized Kaczmarz method applied to the reduced system ATxT=r, where AT is a subset of the columns of A at which the sequence of sets Tj converges. The SRK method is capable of determining sparse solutions to both over and under-determined linear systems, and enjoys faster convergence compared to the randomized Kaczmarz method.
Reducing clutter produced by the wall is described for a number of prior methods. Some methods assume a perfect knowledge of the reflective geometry of the scene. For example, Setlur et al. al. developed multi-path signal models to associate multi-path ghosts to the true locations of the targets, thereby improving the radar system performance by reducing false positives in an original synthetic aperature radar (SAR) image, see Setlur et al., “Multipath model and exploitation in through-the-wall and urban radar sensing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 49, no. 10, pp. 4021-4034,2011.
One method describes a physics based approach to multi-path exploitation where the imaging kernel of the back-projection method is designed to focus on specific propagation paths of interest, see Chang, “Physics-Based Inverse Processing and Multi-path Exploitation for Through-Wall Radar Imaging,” Ph.D. thesis, Ohio State University, 2011.
Another method combines target sparsity with multi-path modeling to achieve further improvements in the quality of TWI, see Leigsnering et al., “Multipath exploitation in through-the-wall radar imaging using sparse reconstruction,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 2, pp. 920-939, April 2014. Specifically, their approach incorporates sources of multi-path reflections of interest into a sparsifying dictionary and solves a group sparse recovery problem to locate the targets from randomly subsampled, frequency stepped SAR data.
TWI can be formulated as a blind sparse-recovery problem, where scene parameters are unknown. Mansour et al. describe multipath-elimination by a sparse inversion (MESI) algorithm that removes the clutter by iteratively recovering the primary impulse responses of targets followed by estimation of corresponding convolution operators that result in multi-path reflections in the received data, see Mansour et al., “Blind multi-path elimination by sparse inversion in through-the-wall-imaging,” Proc. IEEE 5th Int. Workshop on Computational Advances in Multi-Sensor Adaptive Process. (CAMSAP), St. Martin, Dec. 15-18, 2013, pp. 256-259.